in terms of the energy per nucleon \(E_0(\rho )\equiv E(\rho ,0)\)  in symmetric nuclear matter (SNM), the (quadratic) symmetry energy \(E_{\mathrm {sym}}(\rho )\equiv 2^{-1}\partial ^2E(\rho ,\delta )/\partial \delta ^2|_{\delta =0}\) , the fourth-order (quartic) symmetry energy \(E_{\mathrm {sym,4}}(\rho )\equiv {24}^{-1}\partial ^4E(\rho ,\delta )/\partial \delta ^4|_{\delta =0}\) , etc.
If we truncate the above expansion to order \(\delta ^2\) , the resulting EOS is often called the parabolic approximation [1]}.
Moreover, the EOS of SNM, the symmetry energy as well as the fourth-order symmetry energy could be further expanded around the saturation density \(\rho _0\)  as
\(E_0(\rho )\approx & E_0(\rho _0)+\frac{1}{2}K_0\chi ^2+\frac{1}{6}J_0\chi ^3+\cdots ,~~\\E_{\mathrm {sym}}(\rho )\approx &E_{\mathrm {sym}}(\rho _0)+L\chi +\frac{1}{2}K_{\mathrm {sym}}\chi ^2+\frac{1}{6}J_{\mathrm {sym}}\chi ^3+\cdots ,\\E_{\mathrm {sym,4}}(\rho )\approx &E_{\mathrm {sym,4}}(\rho _0)+L_{\mathrm {sym,4}}\chi +\frac{1}{2}K_{\mathrm {sym,4}}\chi ^2+\frac{1}{6}J_{\mathrm {sym,4}}\chi ^3+\cdots ,\) 
